The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels

A connection between the zeta functions of zeros and poles of a meromorphic function has been established, and using it, a criterion for the absence of zeros has been derived. Sufficient conditions for the existence of zeros of sums of Cauchy kernels have been obtained, including those based on geometric conditions on the poles.